IPSec Working Group P. Panjwani and Y. Poeluev
INTERNET-DRAFT Certicom Corp
Expires November 20, 1999 May 26, 1999
Additional ECC Groups For IKE
<draft-ietf-ipsec-ike-ecc-groups-00.txt>
Status of this Memo
This document is an Internet-Draft and is in full conformance with
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Abstract
This document describes new ECC groups for use in IKE [RFC2409] in
addition to the Oakley groups included in RFC 2409. These groups
are defined to align with other ECC implementations and standards,
and in addition, some of them provide higher strength than the Oakley
groups. It should be noted that this document is not self-contained.
It uses the notations and definitions of [RFC2409].
Table of Contents
1. Introduction ............................................... 2
2. Additional Oakley Groups ................................... 3
2.1. Fifth Group .............................................. 3
2.2. Sixth Group .............................................. 4
2.3. Seventh Group ............................................ 5
2.4. Eighth Group ............................................. 6
3. Security Considerations .................................... 7
4. Patent Statements .......................................... 7
5. Acknowledgments ............................................ 7
6. References ................................................. 8
7. Authors' Addresses ......................................... 8
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1. Introduction
This document describes default groups for use in IKE [RFC2409] in addition
to the Oakley groups included in RFC 2409. The document assumes that the
reader is familiar with the IKE protocol, and the concept of Oakley Groups,
as defined in RFC 2409.
RFC 2409 defines four standard Oakley Groups - two modular exponentiation
groups and two elliptic curve groups over GF[2^N]. One modular exponentia-
tion group (768 bits - Oakley Group 1) is mandatory for all implementations
to support, while other three are optional. Both elliptic curve groups
(Oakley Groups 3 and 4) are defined over GF[2^N] with N composite.
Implementations have shown that use of elliptic curve groups can signifi-
cantly improve performance over using Oakley Groups 1 and 2. The purpose
of this document is to expand the options available to implementers of
elliptic curve groups by adding four new groups. The reasons for addition
of these new groups include the following:
- The groups proposed encourage alignment with other elliptic curve
standards. Oakley Groups 3 and 4 were defined prior to availability of
other elliptic curve standards, and they are therefore not aligned with
other efforts. Specifically, unlike Oakley groups 3 and 4, the proposed
groups use base points whose order is prime as required by IEEE [P1363]
and ANSI [X9.62, X9.63], and base points whose prime order is greater
than 2^160, as required by ANSI [X9.62, X9.63].
- Two of the new groups proposed offer higher strength than the existing
Oakley Groups. As computing power increases and other standards such as
the AES are specified it becomes increasingly desirable to make higher
strength groups available to implementers.
- The four groups proposed in this document use elliptic curves over
GF[2^N] with N prime unlike the existing Oakley Groups. This addresses
concerns expressed by many experts regarding curves defined over GF[2^N]
with N composite. It also aligns the groups with plans recently announced
by NIST. NIST have indicated that they will only support curves over
GF[2^N] when the curves over GF[2^N] have N prime.
(It may also be desirable to represent points in the form specified in
IEEE [P1363] and ANSI [X9.62, X9.63] in the key exchange payload
instead of sending only the x-coordinate as currently specified in
[RFC2409]. Since it is unclear exactly how use of a variable length
key exchange payload affects IKE, this has not been suggested at this
time.)
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These groups could also be defined using the New Group Mode, but including
them in this RFC will encourage interoperability of IKE implementations
based upon elliptic curve groups. This is particularly critical, since the
available Oakley Groups based on elliptic curves are insufficient for the
reasons mentioned above. In addition, availability of standardized groups
will result in optimizations for a particular curve and fields size as
well as precomputations that could result in faster implementations.
In summary, due to the performance advantages of elliptic curve groups in
IKE implementations and the need for standardized groups as alternatives
to Oakley Groups 3 and 4, this document defines four new groups based on
elliptic curve groups. The groups are defined at two field sizes: GF[2^163]
and GF[2^277]. These field sizes correspond to 80-bit and 128-bit symmetric
key strengths and 1,024-bit and 3,044-bit Diffie-Hellman respectively. Two
curves are defined at each strength - a Koblitz curve that enables espe-
cially efficient implementations due to the special structure of the curve
[Kob, NSA], and a curve chosen verifiably at random.
2. Additional Oakley Groups
The notation adopted in [RFC2409] is used below to describe the new Oakley
Groups proposed.
2.1 Fifth Group
IKE implementations SHOULD support a EC2N group with the following charac-
teristics. This group is assigned id 5 (five). The curve is based on the
Galois Field GF[2^163]. The field size is 163. The irreducible polynomial
used to represent the field is:
u^163 + u^7 + u^6 + u^3 + 1.
The equation for the elliptic curve is:
y^2 + xy = x^3 + ax^2 + b.
Specifically the group is defined by the following characteristics:
Field size:
163
Irreducible polynomial:
0x0800000000000000000000000000000000000000C9
Group Curve a:
0x07B6882CAAEFA84F9554FF8428BD88E246D2782AE2
Group Curve b:
0x0713612DCDDCB40AAB946BDA29CA91F73AF958AFD9
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Group Generator One:
0x0369979697AB43897789566789567F787A7876A654
Group Generator Two:
0x00435EDB42EFAFB2989D51FEFCE3C80988F41FF883
Group Order:
0x07FFFFFFFFFFFFFFFFFFFE91556D1385394E204F36
The order of the base point P defined by Group Generator One and Group
Generator Two is the prime:
0x03FFFFFFFFFFFFFFFFFFFF48AAB689C29CA710279B
The group order is twice this prime.
The group was chosen verifiably at random using SHA-1 as specified in
[X9.62] from the seed:
0x24B7B137C8A14D696E6768756151756FD0DA2E5C
However, for historical reasons, the method to generate the group from the
seed differs slightly from the method described in [X9.62]. Specifically
the coefficient Group Generator Two produced from the seed is the reverse
of the coefficient that would have been produced by the method described
in [X9.62].
The data in the KE payload when using this group is the value x from the
solution (x,y), the point on the curve chosen by taking the randomly
chosen secret Ka and computing Ka*P, where * is the repetition of the
group addition and double operations, P is the curve point with x-coor-
dinate equal to Group Generator One and y-coordinate equal to Group
Generator Two. This is identical to the method used by Oakley Groups
3 and 4.
2.2 Sixth Group
IKE implementations SHOULD support a EC2N group with the following charac-
teristics. This group is assigned id 6 (six). The curve is based on the
Galois Field GF[2^163]. The field size is 163. The irreducible polynomial
used to represent the field is:
u^163 + u^7 + u^6 + u^3 + 1.
The equation for the elliptic curve is:
y^2 + xy = x^3 + ax^2 + b.
Specifically the group is defined by the following characteristics:
Field size:
163
Irreducible polynomial:
0x0800000000000000000000000000000000000000C9
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Group Curve a:
0x000000000000000000000000000000000000000001
Group Curve b:
0x000000000000000000000000000000000000000001
Group Generator One:
0x02FE13C0537BBC11ACAA07D793DE4E6D5E5C94EEE8
Group Generator Two:
0x0289070FB05D38FF58321F2E800536D538CCDAA3D9
Group Order:
0x0800000000000000000004021145C1981B33F14BDE
The order of the base point P defined by Group Generator One and Group
Generator Two is the prime:
0x04000000000000000000020108A2E0CC0D99F8A5EF
The group order is twice this prime.
The data in the KE payload when using this group identical to the data
used with Oakley Groups 3, 4, and 5.
2.3 Seventh Group
IKE implementations SHOULD support a EC2N group with the following charac-
teristics. This group is assigned id 7 (seven). The curve is based on the
Galois Field GF[2^277]. The field size is 277. The irreducible polynomial
used to represent the field is:
u^277 + u^12 + u^6 + u^3 + 1.
The equation for the elliptic curve is:
y^2 + xy = x^3 + ax^2 + b.
Specifically the group is defined by the following characteristics:
Field size:
277
Irreducible polynomial:
0x2000000000000000000000000000000000000000000000000000000000000000001049
Group Curve a:
0x1853044E52AC1959E666EB976840794626756389C3084E1C0E8EE58B5ADE55B0E94F06
Group Curve b:
0x12709B9501DBD0C98DC5E7E17AF396B445303DFDBDEA0AAE05840A8204625E0B9157B9
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Group Generator One:
0x180949B3BBF7F5168DA7647F9BBAE716F02F6174EC79DE0A5AC9AEC5FF48E4D696323B
Group Generator Two:
0x1CB7297D452004A0F2C34F33E5A6900122103B5F78BE5B838AA97848CCFEDD01F60618
Group Order:
0x1FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF7049860E759CB2BDBEF59DD8C6B43EAE816
The group was chosen verifiably at random using SHA-1 as specified in
[X9.62] from the seed:
0xAC2F14783E695F34335EB4D696E6768756151753
The order of the base point P defined by Group Generator One and Group
Generator Two is the prime:
0x0FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFB824C3073ACE595EDF7ACEEC635A1F5740B
The group order is twice this prime.
The data in the KE payload when using this group identical to the data used
with Oakley Groups 3, 4, 5, and 6.
2.4 Eighth Group
IKE implementations SHOULD support a EC2N group with the following charac-
teristics. This group is assigned id 7 (seven). The curve is based on the
Galois Field GF[2^277]. The field size is 277. The irreducible polynomial
used to represent the field is:
u^277 + u^12 + u^6 + u^3 + 1.
The equation for the elliptic curve is:
y^2 + xy = x^3 + ax^2 + b.
Specifically the group is defined by the following characteristics:
Field size:
277
Irreducible polynomial:
0x2000000000000000000000000000000000000000000000000000000000000000001049
Group Curve a:
0x0000000000000000000000000000000000000000000000000000000000000000000000
Group Curve b:
0x0000000000000000000000000000000000000000000000000000000000000000000001
Group Generator One:
0x1F548FD1F2A95B49A515F99E1933746460B57E47C1AF27AC3E101A1C175C92A741061A
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Group Generator Two:
0x070B258D9BE112C22B9BAA56BBBA6BB9CA38BC0F5E7E95BFD65FBBBC64BC3317DAF873
Group Order:
0x1FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFB42A2D15E3F4D2F69828D921E5BB03C3EEC
The order of the base point P defined by Group Generator One and Group
Generator Two is the prime:
0x07FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFED0A8B4578FD34BDA60A3648796EC0F0FBB
The group order is four times this prime.
The data in the KE payload when using this group identical to the data used
with Oakley Groups 3, 4, 5, 6, and 7.
3. Security Considerations
Since this document proposes new groups for use within IKE, many of the
security considerations contained within RFC 2409 apply here as well.
Two of the groups proposed in this document (seventh and eighth groups)
offer higher strength than those proposed in RFC 2409, since they are
defined over field size of 277 bits. In addition, since all the new
groups are defined over GF[2^N] with N prime, they address concerns
expressed regarding elliptic curve groups included in RFC 2409, which
are curves defined over GF[2^N] with N composite.
4. Patent Statements
To be provided.
[NOTE: The readers should be aware of the possibility that implementation
of this draft may require use of inventions covered by patent
rights.]
5. Acknowledgments
The authors would like to thank Simon Blake-Wilson (Certicom Corp.),
editor for ANSI X9.63 [X9.63], for his comments and recommendations.
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6. References
[RFC2409] Harkins, D. and Carrel, D., The Internet Key Exchange (RFC 2409).
November, 1998.
[X9.62] American National Standards Institute. ANSI X9.62-1999, Public Key
Cryptography for the Financial Services Industry: The Elliptic Curve Digital
Signature Algorithm. January, 1999.
[X9.63] American National Standards Institute. ANSI X9.63-199x, Public Key
Cryptography for the Financial Services Industry: Key Agreement and Key
Transport using Elliptic Curve Cryptography. Working Draft. January, 1999.
[P1363] Institute of Electrical and Electronics Engineers. IEEE P1363,
Standard for Public Key Cryptography. IEEE Microporcessor Standards
Committee. Working Draft. September 1998.
[Kob] Koblitz, N., CM curves with good cryptographic properties.
Proceedings of Crypto '91. Pages 279-287. Springer-Verlag. 1992.
[NSA] Solinas, J., An improved algorithm for arithmetic on a
family of elliptic curves. Proceedings of Crypto '97.
Pages 357-371. Springer-Verlag. 1997.
7. Authors' Addresses
Authors:
Prakash Panjwani
Certicom Corp.
ppanjwani@certicom.com
Yuri Poeluev
Certicom Corp.
ypoeluev@certicom.com
Panjwani and Poeluev [Page 8]
